aa r X i v : . [ h e p - t h ] J a n UTTG-25-2020
On the Development of Effective Field Theory
Steven Weinberg ∗ Theory Group, Department of Physics, University of TexasAustin, TX, 78712
Abstract
This is a lightly edited version of the talk given on September 30, 2020to inaugurate the international seminar series
All Things EFT . It reviewssome of the early history of effective field theories, and concludes with adiscussion of the implications of effective field theory for future discovery.What is the world made of? This question is perhaps is the deepestand earliest in all of science. Greeks were asking this question a hundredyears before the time of Socrates. By the time that I became a graduatestudent an answer had apparently been settled. The world is made not ofwater, earth, air or fire, but of fields. There is the electromagnetic fieldthat when quantum mechanics is applied to it is manifested in the form ofbundles of energy, momentum – particles that are called photons. There isan electron field that similarly when quantized appears as particles calledelectrons. And there are other fields that we in the late 1950s knew wedid not yet know about. The weak and the strong interactions were prettymysterious. It was clear that there had to be more than just electrons andphotons. But we looked forward to a description of nature as consistingfundamentally of fields as the constituents of everything.The quantum field theory of electrons and photons in the late 1940shad scored a tremendous success. Theorists – Feynman, Schwinger, Tomon-aga, Dyson – had figured out after decades of effort how to do calculationspreserving not only Lorentz invariance but also the appearance of Lorentzinvariance at every stage of the calculation. This allowed them to sort outthe infinities in the theory that had been noticed in the early 1930s byOppenheimer and Waller, and that had been the bˆete noire of theoreticalphysics throughout the 1930s. They were able to show in the late 1940s that ∗ Electronic address: [email protected] e / π is 1 / W and Z fields. There were more fermions, not just the electron but a whole host ofcharged leptons and neutrinos and quarks. But the Standard Model seemedto be quantum electrodynamics writ large. One could perhaps have beenforgiven for reaching a stage of satisfaction that, although not everythingwas answered, although there were still outstanding questions, that this wasgoing to be part of nature at the most fundamental level.Now that has changed. In the decades since the completion of the Stan-dard Model a new and cooler view has become widespread. The StandardModel, we now see – we being, let me say, me and a lot of other people – asa low-energy approximation to a fundamental theory about which we knowvery little. And low energy means energies much less than some extremelyhigh energy scale 10 − GeV . As a low energy approximation we ex-pect corrections to the Standard Model. These corrections are beginning to2how up. Some of them have already been found.This whole point of view goes by the name of effective field theory. It’shad applications outside elementary particle physics in areas like supercon-ductivity. I am not going to in this talk try to bring the subject up-to-date,including all the applications of effective field theory to hadronic physics, andto areas of physics outside particle physics, like superconductivity. That’sgoing to be done by subsequent lecturers in this series by physicists whoplayed a leading role in the development of effective field theory beyondanything that I knew about it in the early days. They are true experts inthe field. I won’t dare to try to anticipate what they will say. I’ll talk abouta subject on which I am undoubtedly the world’s expert and that is my ownhistory of how I came to think about these things. I’m a little bit unhappythat I am putting myself too much forward. Other people came to effectivefield theories through different routes. I’m not going to survey anyone else’sintellectual history except my own.From my point of view, it started with current algebra. The late 1950sand early 1960s were a time of despair about the future – about the practi-cal application of quantum field theory to the strong interactions. Althoughwe could believe that quantum field theory was at the root of things wedidn’t know how to apply quantum field theory to the strong interactions.Perturbation theory didn’t work. Instead, a method was developed calledcurrent algebra in which one concentrated on the currents of the weak in-teractions, the vector and axial vector currents, using their commutationrelations, their conservation properties and in particular a suggestion madeby Nambu that the divergence of the axial vector current was dominatedby one pion states. This current algebra was used in a very clunky way, re-quiring very detailed calculations that just called out for a simpler approachto derive useful results, including the Goldberger-Treiman formula for thepion decay amplitude and the Adler-Weisberger sum rule for the axial vectorcoupling constant.After a while some of us began to think that although these resultswere important and valuable, perhaps we were giving too much attentionto the currents themselves, which of course play a central role in the weakinteractions. We ought to concentrate on the symmetry properties of thestrong interactions which made all this possible. In particular, the existenceof a symmetry which gradually emerged in our thinking, chiral SU (2) × SU (2), which is just isotopic spin symmetry applied separately to the lefthanded and right handed parts of what we would now say are quark fields.This symmetry is a property of the strong interactions which would be3mportant even if there weren’t any weak interactions. And it was employedto derive purely strong interaction results, like for example, the scatteringlengths of pions on nucleons and pions on pions and more complicated thingslike the emission of any number of soft pions in high energy collisions ofnucleons or other particles. When this was done using these, as I said, clunkymethods of current algebra, looking at the results, they seemed to look likethe results of a field theory. You could write down Feynman diagrams justout of the blue which would reproduce the results of current algebra.And so the question naturally arose, is there a way of avoiding the ma-chinery of current algebra by just writing down a field theory that wouldautomatically produce the same results with much greater ease and perhapsphysical clarity? Because after all in using current algebra one had to alwayswave one’s hands and make assumptions about the smoothness of matrixelements, whereas if you could get these results from Feynman diagramsyou could see what the singularity structure of the matrix elements was andmake only those smoothness assumptions that were consistent with that.At the beginning this was done using a standard theory with the chiralsymmetry that we thought was at the bottom of all these results, the linearsigma model, and then re-defining the fields in such a way that the resultswould look like current algebra. The effect of the redefinition was the in-troduction of a non-linearly-realized chiral symmetry. Eventually the linearsigma model was scrapped; instead the procedure was simply to ask, whatkind of symmetry transformation for the pion field alone, some transforma-tion into a nonlinear function of the pion field, would have the algebraicproperties of chiral symmetry, based on the Lie algebra of SU (2) × SU (2).That theory had the property that in lowest order in the coupling con-stant 1 /F π the results reproduced the results of current algebra. Why didit? Well, it had to, because a theory having chiral and Lorentz invarianceand unitarity and smoothness satisfied the assumptions of current algegraand therefore had to reproduce the same results. For example current alge-bra calculations gave a pion-pion scattering matrix element of order 1 /F π ,where F π is the pion decay amplitude, so if you use this field theory andjust threw away everything except the leading term which is of order 1 /F π ,this matrix element had to agree with the results of current algebra.In this way phenomenological Lagrangians were developed that couldbe thought of as merely labor-saving devices, which were guaranteed togive the same results as current algebras because they satisfied the sameunderlying conditions of symmetry and so on, and that could be used inlowest order because current algebra said the result was of lowest order in4he 1 /F π coupling – that is, 1 /F π for ππ scattering and also for pion-nucleonscattering. If you had more pions you would have more powers of 1 /F π . Butthe results would always agree with current algebra.No one took these theories seriously as true quantum field theories atthe time. (I am talking about the late 1960s.) No one would have dreamedat this point of using the phenomenological Lagrangian in calculating loopdiagrams. What would be the point? We knew that it was the tree approx-imation that reproduced current algebra. As I said, these phenomenologicalLagrangians were simply labor-saving devices.The late 1960s and 1970s saw many of us engaged in the development ofthe Standard Model. During this time I wasn’t thinking much about currentalgebra or phenomenological Lagrangians. The soft pion theorems had beensuccessful, not only in agreeing with experiment, but also in killing off acompetitor of quantum field theory known as S-matrix theory. S-matrixtheory had been the slogan of a school of theoretical physicists headed byGeoff Chew at Berkeley. I had been there at Berkeley in its heyday but hadnever bought on to it.Their idea was that field theory is hopeless. It deals with things we willnever observe like quantum fields. What we should do is just to study thingsthat are observable, like S-matrix elements: apply principles of Lorentz in-variance, analyticity and so on, and get results like dispersion relations thatwe can compare with observation. It was even hoped that stable or unstablecomposite particles like the ρ meson provide the force that produces thesecomposites, so that using this bootstrap mechanism one could actually docalculations. This never really worked as a calculational scheme, but was ex-tremely attractive philosophically because it made do with very little exceptthe most fundamental assumptions, without introducing things like stronglyinteracting fields that we really didn’t know about. But the chiral symme-try results, the soft-pion results, showed that some of the approximationsassumed in using S-matrix theory, like strong pion-pion interactions at lowenergy, just weren’t right. Chiral symmetry provided actual calculations ofprocesses like ππ and π -nucleon scattering, which made the ideas of S-matrixtheory seem unnecessary, attractive as the philosophy was.S-matrix theory had been largely killed off and chiral symmetry had beenput in the books as a success, but we were all involved in applying the ideas ofquantum field theory in the weak and the electromagnetic interactions andthen the strong interactions, building up the Standard Model, which wasbeginning to be very successful experimentally. It was a very happy timefor the interaction between theory and experiment. During this period of5ourse I was teaching. It was in the course of teaching that my point of viewchanged, because in teaching quantum field theory I had to keep confrontingthe question of the motivation for this theory. Why should these studentstake seriously the assumptions we were making, in particular the formalismof writing fields in terms of creation and annihilation operators, with theircommutation relations. Where did this come from? The standard approachwas to take a field theory like Maxwell’s theory and quantize it, using therules of canonical quantization. Lo and behold, you turn the crank, and outcome the commutation relations for the operator coefficients of the wavefunctions in the quantum field.I found that hard to sell, especially to myself. Why should you applythe canonical formalism to these fields? The answer that the canonical for-malism had proved useful in celestial mechanics in the 19th century wasn’treally very satisfying. In particular, suppose there was a theory that inother ways was successful but couldn’t be formulated in terms of canonicalquantization, would that bother us? In fact, we have quantum field theorieslike that. They’re not realistic theories. They’re theories in six dimensions,or theories we derive by compactifying six dimensional theories. We havequantum field theories that apparently can’t be given a Lagrangian formal-ism – that can’t be derived using the canonical formalism. So I looked forsome other way of teaching the subject.I fastened on a point of view that is really not that different from S-matrix theory. One starts of course by assuming the rules of quantummechanics as laid down in the 1920s, together with special relativity andthen one makes an additional assumption, the cluster decomposition prin-ciple, whose importance was emphasized to me by a colleague at Berkeley,Eyvind Wichmann, while I was there in the 1960s. The cluster decomposi-tion principle says essentially that unless you make special efforts to producean entangled situation the results of distant experiments are uncorrelated.The results of an experiment at CERN are not affected by the results be-ing obtained by an experiment being done at the same time at Fermilab.The natural way of implementing the cluster decomposition principle is bywriting the Hamiltonian as a sum of products of creation and annihilationoperators with non-singular coefficients. Indeed, this had been done formany years by condensed matter physicists not because they were inter-ested in quantum field theory as a fundamental principle, but in order tosort out the volume dependence of various thermodynamic quantities. Theywere managing this by introducing creation and annihilation operators longbefore I began to teach courses in quantum field theory.6t gradually appeared to me in teaching the subject that although indi-vidual quantum field theories like quantum electrodynamcis certainly havecontent quantum field theory in itself has no content except the princi-ples on which it’s based, namely quantum mechanics, Lorentz invariance,and the cluster decomposition principle, together with whatever other sym-metry principles you may want to invoke, like chiral symmetry and gaugeinvariance.This means that if we think we know what the degrees of freedom are– the particles we have to study – like for example, low energy pions, thatif we write down the most general possible theory involving fields for theseparticles, including all possible interactions consistent with the symmetries,which in this case are Lorentz invariance and chiral SU (2) × SU (2), if wewrite down all possible invariant terms in the Lagrangian, and we work toall orders in perturbation theory, the result can’t be wrong, because it’sjust a way of implementing these principles. It is just giving you the mostgeneral possible matrix element consistent with Lorentz invariance, chiralsymmetry, quantum mechanics, cluster decomposition, and unitarity.Now, this may not sound as if it’s a very useful realization. If you tellsomeone to calculate using the most general possible Lagrangian with anunlimited number of free parameters and calculate to all orders in pertur-bation theory they’re likely to seek some advice elsewhere on how to spendtheir time. But it’s not that bad, because even if the theory has no smalldimensionless couplings, you can use this approach to generate a power se-ries in powers of the energy that you’re interested in. For instance, if you’reinterested in low-energy pions and you don’t want to consider energies highenough so that ππ collisions can produce nucleon-antinucleon pairs, you willbe dealing with typical energies E well below the nucleon mass. This verygeneral Lagrangian gives you a power series in powers of E . Specifically,aside from a term that depends only on the nature of the process being con-sidered, the number of powers of E arising from a given Feynman diagramis the total number of derivatives acting at all the vertices, plus half thenumber of nucleon lines connected to all the vertices, plus twice the numberof loops.Now, chiral symmetry dictates that the number of derivatives plus halfthe number of nucleon fields in each interaction is always equal to or greaterthan two. The diagrams that give the lowest total number of powers of E are those constructed only from interactions where that number equalstwo, with no loops. For pion-pion scattering the leading term comes from asingle vertex with just two derivatives. For pion-nucleon scattering there is7 diagram with a single vertex with one deerivative to which are connectedtwo nucleon lines. Thsee diagrams are the ones that we had been using sincethe mid 1960s to reproduce the results of current algebra.But now the new thing was that you could consider contributions ofhigher order in energy. If you look for terms that have two additional pow-ers of energy they could come from diagrams where you have one interactionwith the number of derivatives plus half the number of nucleon fields equal-ing not two but four, plus any number of interactions with this numberequal to two, and no loops. Or you could have only interactions with thenumbers of derivatives plus half the number of nucleon fields equal to two,that is, just the basic interactions that reproduce current algebra, plus oneloop. The infinity in the one loop diagram could be canceled by that oneadditional vertex that has, say, not two derivatives but four derivatives. Inevery order of perturbation theory as you encounter more and more loops,because you are allowing more and more powers of energy, you get moreand more infinities, but there are always counterterms available to cancelthe infinities. A non-renormalizable theory, like the soft pion theory, is justas renormalizable as a renormalizable theory. You have an infinite numberof terms in the Lagrangian, but only a finite number are needed to calculateS-matrix elements to any given order in E .Similar remarks apply to gravitation, which I think has led to a newperspective on general relativity. Why in the world should anyone takeseriously Einstein’s original theory, with just the Einstein-Hilbert action inwhich only two derivatives act on metric fields? Surely that’s just the lowestorder term in an infinite series of terms with more and more derivatives. Insuch a theory, loops are made finite by counterterms provided by the higherorder terms in the Lagrangian. This point of view has been actively pursuedby Donoghue and his collaborators.Teaching came to my aid again. At the blackboard one day in 1990 itsuddenly occurred to me that one of the kinds of interaction that has thenumber of derivatives plus half the number of nucleon fields equal to twois the interaction with no derivatives at all and four nucleon fields. It hadtaken me a decade to realize that four divided by two is two. This sortof interaction is just the kind of hard-core nucleon-nucleon interaction thatnuclear physicists had always known would be needed to understand nuclearforces. But now we had a rationale for it. In a calculation of nuclear forcesas a power series in energy, the leading terms are just the ones that thenuclear physicists had always been using, pion exchange plus a hard core.This point of view has been explored by Ordo˜nez and van Kolck and others.8n the theories that I have been discussing, the chiral symmetry theory ofsoft pions and general relativity, the symmetries don’t allow a purely renor-malizable interaction. In a theory of gauge fields and quarks and leptonsand scalars you can have a renormalizable Lagrangian. Throwing away ev-erything but renormalizable interactions you have just the Standard Model,a renormalizable theory of quarks and leptons and gauge fields and a scalardoublet. But now we can look at the Standard Model as one term in a muchmore general theory, with all kinds of non-renormalizable interactions, whichyield corrections if higher order in energy.In these corrections the typical energy E must be divided by some char-acteristic mass scale M . For the chiral theory of soft pions this mass scaleis M ≈ πF π , about 1200 MeV. For the theory of weak, strong and electro-magnetic interactions M is probably a much higher scale, perhaps somethinglike the scale of order 10 GeV where Georgi, Quinn and I found the effec-tive gauge couplings of the weak, strong, and electromagnetic interactionsall come together. Or perhaps it’s the characteristic scale of gravitation, thePlanck scale 10 GeV . Or perhaps somewhere in that general neighbor-hood. It is the large scale of M that made it a good strategy in constuctingthe Standard Model to look for a renaormalizable theory .We now expect that there are corrections to the renormalizable StandardModel of the order of powers of E/M . How in the world are we ever goingto find corrections that are suppressed by such incredibly tiny fractions?The one hope is that those corrections can violate symmetries that we oncethought were inviolable, but that we now understand are simply accidents,arising from the constraint of renormalizability that we imposed on theStandard Model.Indeed, quite apart from the development of effective field theory, oneof the great things about the Standard Model was that it explained vari-ous symmetries that could not be fundamental, becasue we already knewthey were only partial or approximate symmetries. This included flavorconservation, such as strangeness conservation, a symmetry of the strongand electromagnetic interactions that was manifestly violated in the weakinteractions. Another example was charge conjugation invariance – likewisea good symmetry of strong and electromagnetic but violated by weak in-teractions. The same was true of parity, although in this case you have tomake special accommodations for non-perturbative effects. All thse wereaccidental symmetries. imposed by the simplicity of the Standard Modelnecessary for renormalizability plus other symmetries like gauge symmetriesand Lorentz invariance that seem truly exact and fundamental. Chiral sym-9etry itself is such an accidental symmetry, though only approximate. Itbecomes an accidental exact symmetry of the strong interactions in the limitin which the up and down quark masses are zero, as does isotopic spin sym-metry. Since these masses are not zero but relatively small chiral symmetryis an approximate accidental symmetry.Now, coming back to effective field theory, there are other symmetrieswithin the Standard Model that are accidental symmetries of the wholerenormalizable theory of weak, strong, and electromagnetic interactions: inparticular baryon conservation and lepton conservation are respected asidefrom very small non-perturbative effects (well, very small at least in labo-ratories, though maybe not so small cosmologically). If baryon and leptonconservation are only accidental properties of the Standard Model maybethey are not symmetries of nature. In this case there is no reason whybaryon and lepton conservation should be respected by non-renormalizablecorrections to the Lagrangian, and so you would expect terms of O ( E/M )or O (( E/M ) ) or higher order as corrections to the Standard Model thatviolate these symmetries.Wilczek and Zee and I independently did a catalog of the leading terms ofthis type. Some of them – those involving baryon number non-conservation– give you corrections of O (( E/M ) ). They have not been yet been discov-ered experimentally. But there are other terms that produce corrections of O ( E/M ) that violate lepton conservation, and they apparently have beendiscovered, in the form of neutrino masses. I wish I could say that the ef-fective field theory point of view had predicted the neutrino masses. Unfor-tunately it must be admmitted that neutrino masses were already proposedby Pontecorvo as a solution to the apparent deficit of neutrinos coming fromthe Sun. So though I can’t say that the effective field theory approach hadpredicted neutrino masses, I do think that it gets a strong boost from thefact that we now have evidence of a correction to the renormalizable part ofthe Standard Model. But of course there already was a known correction.Gravitation was always there, warning us that renormalizable quantum fieldtheory can’t be the whole story.I expect that sooner or later we will be seeing another departure from therenormalizable Standard Model in the discovery of proton decay, or someother example of baryon nonconservation. In a sense baryon nonconservationhas already been discovered, because we know from the present ratio ofbaryon number to photon number that in the early universe before thetemperature dropped to a few GeV there was about 1 excess quark overevery 10 quark-antiquark pairs. This has to be explained by the non-10enormalizable corrections to the Standard Model, and indeed it has beenexplained, unfortunately not just by one such model but by many differentmodels. We don’t know the actual mechanism for producing baryon numberin the early universe, but I have no doubt that it will be found.There are still unnatural elements in the Standard Model. I said thatyou would expect the leading terms that describe physical reactions to begiven by the renormalizable theory – here the Standard Model – with theeffects of non-renormalizable terms suppressed by powers of E/M . Thosecorrections come from interactions that have coupling constants whose di-mensions have negative powers of mass, 1 /M , 1 /M , and so on. But whatabout interactions that have positive powers of mass? Why aren’t they thereat O ( M n )? Well unfortunately we don’t have a good explanation.The cosmological constant is such a term. It has the dimensions ofenergy per volume, in other words M . We don’t know why it’s as small asit is. There is also the bare mass of the Higgs boson. That’s the one term inthe Standard Model Lagrangian whose coefficient has the dimensionality of apositive power of mass. We don’t know why it is not O (10 GeV ). These aregreat mysteries that confront us: Why are the terms in our present theorythat have the dimensions of positive powers of mass so small compared to thescale that we think is the fundamental scale, somewhere in the neighborhoodof 10 − GeV ? We don’t know.With the new approach to the Standard Model I think we have to saythat this theory in its original form is not what we thought it was. It’s not afundamental theory. But at the same time I want to stress that the StandardModel will survive in any future textbooks of physics in the same way thatNewtonian mechanics has survived as a theory we use all the time appliedto the solar system. All of our successful theories survive as approximationsto a future theory.There’s a school of philosophy of science associated in particular with thename of Thomas Kuhn, that sees the development of science – particularly ofphysics – as a series of paradigm shifts in which our point of view changes soradically that we can barely understand the theories of earlier times. I don’tbelieve it for a minute. I think our successful theories always survive, asNewtonian mechanics has, and as I’m sure the Standard Model will survive,as approximations.Now we have to face the question, approximations to what? We thinkthe Standard Model is a low energy approximations to a theory whose con-stituents involve mass scales way beyond what we can approach in the labo-ratory, scales on the order of 10 , 10 GeV . It may be a field theory. It may11e an asymptomatically safe field theory, which avoids coupling constantsrunning off to infinity as the energy increases. Or it seems to me more likelythat it’s not a field theory at all, that it’s something like a string theory.In this case we will understand the very successful field theories with whichwe work as effective field theories, embodying the principles of quantummechanics and symmetry, applied as an approximation valid at low energywhere any theory will look like a quantum field theory.